Introduction

• Purchasing power is ‘the value of money, thought of as how much it can buy’ (Merriam Webster)
• Maintaining purchasing power is integral to material living standards, as a decrease in purchasing power means that for the same nominal cost, you are unable to access the same basket of goods and services
• Purchasing Power Parity theory tells us that in the long run, prices should be as such that taking into account the foreign exchange rate you should have the same purchasing power in both countries (UBC, 2016)
• For example in the long run, it should matter if you buy a Big Mac in Australia, or anywhere else in the world, the price adjusted for the exchange rate should be the same
• This is how the Big Mac Index was born in 1986 by the Economist Magazine (The Economist, 2017) as an informal way to show the purchasing power of a currency
• It uses the price of a McDonald’s Big Mac burger to compare purchasing power across different countries as the product is directly comparable and is sold globally

Problem Statement

• Today we are testing for purchasing power parity, using the Big Mac Index. The Economist uses the American dollar as it’s baseline currency, however to test our hypothesis, we will use the Australian dollar as the baseline currency, our hypothesis testing is done from an AUstralian perspective.

• The purchasing power parity theory holds when there is no difference in purchasing power between countries, which can be written as: \[ FXrate = ForeignBigMacPrice/LocalBigMacPrice \]

• We do not expect purchasing power parity theory to hold in the short run due to price stickiness, which is a resistance to price changes even when broader macro-economic factors suggest that this price isn’t optimal (Investopedia)

Data

• The data is a compilation; the official daily foreign exchange rates were sourced from the Reserve Bank of Australia (2017), there was 859 observations from January 2014 - June 2017 for each foreign exchange rate. Due to the Reserve Bank of Australia being a reputable source, no assumptions have been made about the data
• The McDonald’s prices were sourced from the Economist (2017), there is an assumption that the as the Economist only records price changes every six months when they occur, as some years there is no price change, that is an accurate gauge
• The Observed Big Mac Index is then calculated by dividing the foreign Big Mac price by the local Big Mac price, this is directly comparable with the foreign exchange rate for that period as it has the same scale \[ Observced Big Mac Index = ForeignBigMacPrice/LocalBigMacPrice \]

Data Cont.

The variables used in analysis are as follows, paired by country: - Foreign Exchange Rate (USD/AUD) - United States of America - Big Mac Index (USD/AUD) - United States of America - Foreign Exchange Rate (YUAN/AUD) - China - Big Mac Index (YUAN/AUD) - China - Foreign Exchange Rate (GBP/AUD) - United Kingdom - Big Mac Index (GBP/AUD) - United Kingdom - Foreign Exchange Rate (NZD/AUD) - New Zealand - Big Mac Index (NZD/AUD) - New Zealand These are all scaled by the foreign country over the Australian Dollar

• The countries chosen for this experiment are the United States, the home of the McDonald’s corporation and the world’s largest economy (CNN Money), China which is Australia’s top export market (Austrade), Great Britain due to its historical ties, and New Zealand as Australia’s closest neighbour.

Descriptive Statistics and Visualisation - Big Mac Index (United States of America)

• Visually, the box plots shows far more variability in the exchange rate than the Big Mac Index, this is to be expected due to the price stickiness, however the interquartile range of the Big Mac Index does not overlap the interquartile range of the exchange rate.
• The median US/AUD comparatively also look to be much higher when using the exchange rate to the index
• Similarly looking at the mean and median summary of the difference, this isn’t zero or very close to zero which we’d expect in the long run as per the purchasing power theory

Descriptive Statistics and Visualisation - Big Mac Index (China)

• In the boxplot, the quantiles for the two methods don’t overlap, showing a significant difference in the two measures.
• The summary statistics show that the minimum value and maximum value for the difference are both below zero, this means that the Big Mac Index is always smaller than the exchange rate for the period.

Descriptive Statistics and Visualisation - Big Mac Index (United Kingdom)

• Although there is greater variability in the visual representation of the exchange rate, the medians appear to be very similar
• This is further shown through the median recorded for the difference being the smallest observed so far

Descriptive Statistics and Visualisation - Big Mac Index (New Zealand)

• The median appears to be higher using the index method in comparison to the exchange rate
• The median in the difference echoes this and the mean is even larger

Hypothesis Testing

• In order to test whether there is a significant difference between the exchange rate and the purchasing power observed by the Big Mac Index, the most appropriate statistical test to use is paired-sample t-test as each Big Mac Index observation should be compared with the exchange rate of the same date period.

• The paired-sample t-test assumes normality, so prior to running the test, there must be visual inspection of the quantile-quantile plot to check this is upheld.

• The null hypothesis is that there is no difference between the exchange rate and the Big Mac Index, so the purchasing power parity theory would hold. \[ H_0: \mu_\triangle=0 \]

• The alternative hypothesis is that there is a significant difference between the exchange rate and the Big Mac Index observed. \[ H_A: \mu_\triangle\neq0 \]

Hypothesis Testing - Big Mac Index (United States of America)

Using the Quantile-Quantile Plot to check the normality of the difference, we can see that there are significant outliers from the normal distribution, so we cannot make inferences from the paired hypothesis test.

## 
##  Paired t-test
## 
## data:  BigMacIndex$`PPPAU/US` and BigMacIndex$FXRUSD
## t = 48.737, df = 858, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.1082509 0.1173357
## sample estimates:
## mean of the differences 
##               0.1127933

Hypothesis Testing - Big Mac Index (China)

Using the Quantile-Quantile Plot to check the normality of the difference, there are outliers at the tails, however for the most part the distribution appears to follow the normal distribution. The paired sample t-test shows a significant mean difference between the exchange rate and index, t(df=858)=-140.67, p-value < 0.05. As the 95% confidence interval [-1.846,-1.795] does not contain zero, we are able to reject the null hypothesis.

## 
##  Paired t-test
## 
## data:  BigMacIndex$`PPPAU/CH` and BigMacIndex$FXRCR
## t = -140.67, df = 858, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -1.846104 -1.795297
## sample estimates:
## mean of the differences 
##                 -1.8207

Hypothesis Testing - Big Mac Index (United Kingdom)

As the paired sample test relies on the assumption of normality and there is many observations that are outliers, we are not same to take inferences from this test.

## 
##  Paired t-test
## 
## data:  BigMacIndex$`PPPAU/UK` and BigMacIndex$FXRUKPS
## t = 2.9083, df = 858, p-value = 0.003728
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.001666402 0.008584493
## sample estimates:
## mean of the differences 
##             0.005125447

Hypothesis Testing - Big Mac Index (New Zealand)

Testing for a significant difference between the exchange rate and index for purchasing power parity, visually the quantile-quantile plot shows that observations appear to be normally distributed. The paired sample t-test shows a significant mean difference between the exchange rate and index, t(df=858) =12.594, p-value < 0.05. As the 95% confidence interval [0.012,0.016] does not contain zero, we can reject the null hypothesis.

## 
##  Paired t-test
## 
## data:  BigMacIndex$`PPPAU/NZ` and BigMacIndex$FXRNZD
## t = 12.594, df = 858, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.01202473 0.01646465
## sample estimates:
## mean of the differences 
##              0.01424469

Discussion

• The major findings of this report are that we could not prove the purchasing power parity theory, from our hypothesis testing, a significant mean difference was found between the Big Mac index and the USD/AUD exchange rate and NZD/AUD exchange rate.
• However, as the differences for the United States and United Kingdom didn’t follow the normal distribution, we were not able to make inferences from any statistal tests performed on these.
• The strength of this report is that a having a goods basket of a McDonald’s Big Mac is the same across different countries due to McDonald’s corporation having global reach. As we aren’t trying to compare the purchasing power of different items, we can have a consistent benchmark.
• Although this is a strength, it does mean that index is at liberty of private pricing and can be influenced by other factors besides from the exchange rate.
• The other issue is that pricing is sticky, which means that prices don’t change day-to-day but only will over the long-term rather than in the short run, but as the Australian exchange is extremely volatile, this changes throughout the day.
• Furthermore, the Economist study took an American perspective using the USD as the base currency, the US dollar is more stable than the Australian. The AUD is known as a cowboy currency due to it being commonly for speculative purposes, which may be why there is greater variability in the data.
• In Australia, as well as many other countries, there is no legal obligation to price a Big Mac the same between stores (McDonalds, 2011). This may mean that the price recorded may not reflect the pricing in all stores.

Discussion Continued…

• Further research needs to be done to look at normalising the data, this may include looking to see whether there is de-trending that needs to completed or if there is a more appropriate statistical test that doesn’t require the data to follow the normal distribution.
• The statistical analysis suggests that the purchasing power parity theory does not apply in the long run, however as the some of the difference didn’t follow the normal distribution further testing is required.

References

• Purchasing Power Definiton, Merriam Webster, retrieved from https://www.merriam-webster.com/dictionary/purchasing%20power
• The Big Mac index - Global exchange rates, 2017, D.H. & R.L.W, The Economist, retrieved from http://www.economist.com/content/big-mac-index
• Purchasing Power Parity, 2016, W.Antweiler, University of British Columbia Sauder School of Business, retrieved from http://fx.sauder.ubc.ca/PPP.html
• Price Stickiness Definition, Investopedia, retrieved from http://www.investopedia.com/terms/p/price_stickiness.asp
• Exchange Rates - Daily - 2014 to Current - F11.1, 2017, RBA, retrieved from http://www.rba.gov.au/statistics/tables/xls-hist/2014-current.xls?v=2017-06-0417-55-54
• Big Mac File, 2017, The Economist, retrieved from http://infographics.economist.com/2017/databank/BMFile2000toJan2017.xls
• World’s Largest Economies, 2017, C.Riley & I.Sherman, CNN Money, retrieved from http://money.cnn.com/news/economy/world_economies_gdp/index.html
• Market Profile - China, Austrade, retrieved from https://www.austrade.gov.au/Australian/Export/Export-markets/Countries/China/Market-profile
• Do All McDonalds Restaurants Charge The Same Price, 2011, McDonalds, retrieved from http://www.mcdonalds.co.uk/ukhome/whatmakesmcdonalds/questions/runningthe-business/prices/do-all-mcdonalds-restaurants-charge-the-same-prices.html